I'm learning about optimal control, and I've got a goofy question. My professor wrote out that the discrete-time form of LQR is given as
$ \text{min}\ J = \frac{1}{2} x_N^TH_Nx_N + {\sum}_{k=0}^{N-1} \frac{1}{2} \{x_k^TQ_kx_k + u_k^TR_ku_k\}$
where $x_{k+1} = A_kx_k + B_ku_k$, $Q_k$ is positive semidefeinite, and $H_N$ and $R_k$ are positive definite.
I learned that the first term is known as the system's terminal cost, while the second term is known as the system's running cost. I see how the sum of these terms would constitute the total system cost, but why can't we lump the terminal cost and running cost together so that
$J = {\sum}_{k=0}^{N} \frac{1}{2} \{x_k^TQ_kx_k + u_k^TR_ku_k\}$
And just minimize that? Any insight would be appreciated! Thanks!
They way you propose is also possible. It just a different way one can define the problem. With your formulation you do have that $u_N$ is obsolete (or trivially would be zero), since it hasn't had time to have any impact on the state of the system (within the horizon of the considered cost function).
It can also be noted that with the given formulation $H_N$ is the boundary condition for associated Riccati difference equation. So denoting it with a different symbol from the other $Q_k$ can help highlights this.